Microfluidics/Capillary effects

Capillary effects are very important at small scales (see previous chapter [Physics_at_smaller_scales]).

They appear at interfaces between a liquid and another material liquid, gas, or solid.

The interfaces give rise to a surface tension that rules the dynamics of this interface.

Creating a new interface costs a work

E=σA,{\displaystyle E=\sigma \,A,}

with σ{\displaystyle \sigma } the surface tension (in N/m) and A{\displaystyle A} the area of the interface. For water/air interface the value of surface tension at 20°C is σ=73×10−3{\displaystyle \scriptstyle \sigma =73\times 10^{-3}}N/m.

This surface tension gives rise to a common phenomenon: liquid droplets tend to become spheres in order to minimize their surface.

where we have introduced the contact angleθ{\displaystyle \theta }. Along the vertical axis we have

σsin⁡θ=N,{\displaystyle \sigma \sin \theta =N,}

N{\displaystyle N} being the reaction from the solid. This shows clearly that the droplet is pulling the surface. This effect can be seen if a droplet is deposited on fresh paint: the contact line will pull a rim that will remain.

The liquid spreads, and form a film coating the surface. It happens when (σsolid−gas−σsolid−liquid)/σ>1{\displaystyle (\sigma _{solid-gas}-\sigma _{solid-liquid})/{\sigma }>1} then σsolid−gas>σsolid−liquid+σ{\displaystyle \sigma _{solid-gas}>\sigma _{solid-liquid}+\sigma }, it costs a lot of energy to have a bare solid.

The liquid droplet pearls, and assumes a perfectly spherical shape. It occurs on surfaces such that (σsolid−gas−σsolid−liquid)/σ<−1{\displaystyle (\sigma _{solid-gas}-\sigma _{solid-liquid})/{\sigma }<-1} then σsolid−liquid>σsolid−gas+σ{\displaystyle \sigma _{solid-liquid}>\sigma _{solid-gas}+\sigma }, the solid does not like to be covered.

These surfaces can be found in nature: observe what happens to a water droplet falling on a lotus leaf. Microscopically these surfaces are covered with little pillars.

A voltage can be applied on a droplet, if the surface is an electrode and an other electrode is in contact with the liquid droplet. A voltage difference V{\displaystyle V} changes the surface energy by:

If only one side of the droplet is sitting on an electrode, the droplet will move towards the electrode, since surface energy is lower there.

Note that electrodes can be insulated to prevent electrolysis of liquids: the technique is called electrowetting on a dielectric (EWOD).

Driving droplets or bubbles in channels acting on surface tension[edit]

It is possible to move droplet or bubbles squeezed in microchannels, even if there liquid is totally wetting the solid. The method is to modify the liquid-gas surface tension (instead of the solid-liquid surface tension).

Attraction of a charged droplet in a microchannel filled with another fluid

If the droplet/bubble is charged on its surface after having been in contact with an electrode, it can be moved afterwards if it is submitted to a voltage gradient. The electric potential of the droplet decreases when charges are closer to the electrode carrying the opposite charge.

Surfactants are molecules that have an affinity with surfaces. They change the surface tension, and are therefore named tensio-active molecules. Usually they are amphiphilic molecules. Amphi- is a Greek prefix, also found in the word amphitheater, meaning dual. Indeed they have both an hydrophilic part and an hydrophobic part. For instance soap molecules have a carboxylic (hydrophilic head) and a long lipidic (hydrophobic chain). At low concentration they all go at the interface, while at high concentration they tend to form micelles.

They lower the surface tension compared to that of water. A free surface has a tension σ{\displaystyle \sigma }=73 mN/m. Surface tension decreases down to a plateau, when the surface is fully covered with surfactant molecules, presenting their lipidic tail to the gas. At this point surface tension is close to that of a lipid surface, around 30 mN/m. The saturation occurs at the critical micellar concentration (CMC), when surface is full and addition of molecules creates new micelles in the liquid.

The inkjet printer involves the formation of tiny droplets of inks. Tiny droplet or bubbles are promising nanoliter containers: they do not have real walls, contain a very small amount of liquid. It is possible to create them, transport them and finally coalesce them with other.

We all have seen the formation of a liquid drop at the exit of a tap. If you slowly open the tap, the droplet volumes increases until its weight is larger than the capillary force. The droplets then detaches and falls. The forces in presence are:

reducing the tap radius does not decrease quickly the drop radius, and droplets are getting very large in front of the tap radius at small scales. We see again that gravity forces become ineffective in front of capillary forces at small scales.

Another strategy, that does not involve gravity, should be considered.

Another geometry is helpful, to produce very calibrated droplet sizes: the fluid A to break is focused by a stream of fluid B through a tiny orifice of size d{\displaystyle d}. Both fluids do not flow simultaneously through the orifice (it is the case only a very large velocities). Break-up involves two steps, when flow rates of fluids are imposed

blocking: when fluid A is in the orifice it makes a plug to fluid B

squeezing: a pressure builds up in fluid B and pinches fluid A until rupture, releasing a droplet

This last step takes a characteristic time τ∼d3/QB{\displaystyle \tau \sim d^{3}/Q_{B}}, with QB{\displaystyle Q_{B}} the flow rate of fluid B. The resulting volume of fluid A inflated during this time gives the droplet volume